 The Planet Mercury's orbit test, Einstein's general relativity theory, actually starts with the planet Uranus, discovered by William Herschel in 1781. By the early 1800s, it was understood that planetary orbits were elliptical with small deviations called perturbations, and that each orbit's closest approach to the Sun, called a perihelion, shifted slightly over each orbit. This is called precession. Using Newton's gravitational equations, all known perturbations and precessions were calculated and found to fit the observations for all the planets except one, Uranus. Careful study over decades showed that Uranus's orbit did not fit Newton's equations. At times, it was moving faster than predicted, and at other times, it was moving slower. There were two schools of thought at that time. One held that Newton's theory did not hold up that far from the Sun, indicating that a new theory was needed. The other proposed that there's another planet beyond Uranus that pulled on it. Uranus's observed deviations could be explained as perturbations. If correct, this would keep Newton's theory intact. Astronomer Urbain Laveret went to work to try discover this new planet. Early in 1846, he published calculations that came very close to where it actually was. On September 23rd of that same year, Johann Gaul, an astronomer at Berlin Observatory, and a student, Heinrich Louis Dearest, found the new planet looking where Laveret had placed it. This planet is now called Neptune. The reason this is relevant for Mercury is that the overall thinking at the time was similar. Newton's theory does not fully explain the observed precession of Mercury's perihelion. In 1859, astronomers, including Laveret, theorized that another planet inside the orbit of Mercury could account for the observations, much like how Neptune explained Uranus's orbital irregularities. The proposed planet between the Sun and Mercury was even given a name, Vulcan. But no such planet was ever observed. Another school of thought held that Newton's theory simply did not hold up that close to the Sun. Einstein was one of them. And his general theory of relativity, describing the impact of the curved space near the Sun, provides a full explanation for the observed precession without the need for an extra planet. Einstein himself thought that this result was the most critical test of his theory. Here's how it works. In 1916, the same year that Einstein published his general relativity paper, Carl Schwarzschild published his exact solution for space around a large, non-rotating mass. His metric is now called a Schwarzschild metric, and it works quite well for slowly rotating masses like the Earth and the Sun and the planets in our solar system. We'll use this metric for the first three tests. As seen from Earth, the precession of Mercury's orbit is measured to be 0.56 arc seconds per orbit. An arc second is 1.36 of a degree, taking into account all the perturbation effects from all the other planets, as well as a very slight deformation of the Sun due to its rotation, and the fact that the Earth is not an inertial frame of reference, Newton's equations predict a precession of 0.5557 arc seconds. That's 0.0043 arc seconds short. With Schwarzschild's metric, Einstein came out with 0.0043 due to the curvature of space near the Sun. This was the exact number to cover the difference. He had passed the first test of his new theory. It's the curved space around the Sun defined by the Schwarzschild metric that produces this small additional precession on each orbit. Here's what it looks like. If we draw the circumference of the Earth's orbit, we get a length that is two pi times our distance from the Sun. If we existed in flat Euclidean space, we would calculate the circumference of an orbit one kilometer closer to the Sun and see that the distance between the orbits is one kilometer. But because of our positive curvature, if we were to measure the circumference with a radius that is one kilometer shorter than the first, we'd find that it is less than two pi times the shorter radius, which means that the distance between the circumferences would be greater than the one kilometer difference in the radii, but only a little. We can repeat this process all the way to the surface of the Sun. With each successive radius, the difference between the orbits would increasingly diverge from the Euclidean numbers. If we were to telescope this picture, you'd see the standard diagrams that are used to help explain general relativity, but diagrams like this are misleading in two ways. First, they represent an external curvature into another dimension, when in fact we are talking about intrinsic curvature. There is no evidence for the existence of a fourth spatial dimension. Second, it looks like you need a downward force on the object to get it to drop into the hole. That would be gravity, but that's what the lines were supposed to represent. So we'll avoid using this technique as we move on to the bending of light by the Sun.